# Obtain all other zeroes of 3x^{4} + 6x^{3} - 2x^{2} - 10x - 5, if two of its zeroes are √5/3 and -√5/3.

**Solution:**

Given polynomial p(x) = 3x^{4} + 6x^{3} - 2x^{2} - 10x - 5

Two zeroes of the polynomial are given as √(5/3) and -√(5/3)

Therefore,

[x - √(5/3)] [x + √(5/3)] = (x^{2} - 5/3) is a factor of 3x^{4} + 6x^{3} - 2x^{2} - 10x - 5.

Therefore, we divide the given polynomial by (x^{2} - 5/3)

Therefore, 3x^{4} + 6x^{3} - 2x^{2} - 10x - 5 = (x^{2} - 5/3)(3x^{2} + 6x + 3) + 0

= 3 (x^{2} - 5/3) (x^{2} + 2x + 1)

On factorizing x^{2} + 2x + 1, we get (x + 1)^{2}

Therefore, its zero is given by

x + 1 = 0

x = - 1

As it has the term (x + 1)^{2},

Therefore, there will be two identical zeroes at x = - 1

Hence the zeroes of the given polynomial are √(5/3) and -√(5/3), - 1 and - 1.

**☛ Check: **NCERT Solutions for Class 10 Maths Chapter 2

**Video Solution:**

## Obtain all other zeroes of 3x⁴ + 6x³ - 2x² - 10x - 5, if two of its zeroes are √5/3 and -√5/3.

NCERT Solutions Class 10 Maths Chapter 2 Exercise 2.3 Question 3

**Summary:**

All the other zeroes of 3x^{4} + 6x^{3} - 2x^{2} - 10x - 5, if two of its zeroes are √5/3 and -√5/3 are -1, and -1.

**☛ Related Questions:**

- Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:(i) p(x) = x3 - 3x2 + 5x - 3, g(x) = x2 - 2(ii) p(x) = x4 - 3x2 + 4x + 5, g(x) = x2 + 1 - x(iii) p(x) = x4 - 5x + 6, g(x) = 2 - x2
- Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:(i) t2 - 3, 2t4 + 3t3 - 2t2 - 9t - 12(ii) x2 + 3x + 1, 3x4 + 5x3 - 7x2 + 2x + 2(iii) x3 - 3x + 1, x5 - 4x3 + x2 + 3x + 1
- On dividing x3 - 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x - 2 and - 2x + 4, respectively. Find g (x).
- Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and (i) deg-p(x) = deg q(x) (ii) deg q(x) = deg r (x) (iii) deg r (x) = 0